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Theorem xmstopn 22664
Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
xmstopn (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn
StepHypRef Expression
1 isms.j . . 3 𝐽 = (TopOpen‘𝐾)
2 isms.x . . 3 𝑋 = (Base‘𝐾)
3 isms.d . . 3 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
41, 2, 3isxms 22660 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
54simprbi 492 1 (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106   × cxp 5353  cres 5357  cfv 6135  Basecbs 16255  distcds 16347  TopOpenctopn 16468  MetOpencmopn 20132  TopSpctps 21144  ∞MetSpcxms 22530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-res 5367  df-iota 6099  df-fv 6143  df-xms 22533
This theorem is referenced by:  imasf1oxms  22702  ressxms  22738  prdsxmslem2  22742  tmsxpsmopn  22750  xmsusp  22782  cmetcusp1  23559  minveclem4a  23636  minveclem4  23638  qqhcn  30633  rrhcn  30639  rrexthaus  30649  dya2icoseg2  30938  sitmcl  31011
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